If x + 1/x = 1 then x^(1729) + x^(-1729) = ?

(X+1/X)^2 = X^2 + 1/X^2 + 2 (X+1/X)^1729 = X^1729 + 1/X^1729 + 2 { 1/X=X^-1 ...... 1/X^1729=X^-1729....} (X+1/X)^1729 = X^1729 + X^-1729 + 2 X^1729 + X^-1729 = (X+1/X)^1729 - 2 X^1729 + X^-1729 = 1^1729 - 2 {X + 1/X = 1...........} X^1729 + X^-1729 = 1-2 X^1729 + X^-1729 = -1

If we know x + 1/x =2 then what would be the value of x^2048 + 1/x^2048 +x^2047 - 1 / x^2047 + 1/x^2049 - x^2049 +2

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x, y, z and k are four non zero positive integers satisfying 1/x + y/2 = z/3 + 4/k, minimum integral value of k for integral value of x, y and z will be